A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers Page: I
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Allen, Cristian G. A Classification of the Homogeneity of Countable Products of
Subsets of Real Numbers. Doctor of Philosophy (Mathematics), August 2017, 75 pp., 26
Spaces such as the closed interval [0, 1] do not have the property of being
homogeneous, strongly locally homogeneous (SLH) or countable dense homogeneous
(CDH), but the Hilbert cube has all three properties. We investigate subsets X of real
numbers to determine when their countable product is homogeneous, SLH, or CDH.
We give necessary and sufficient conditions for the product to be homogeneous. We
also prove that the product is SLH if and only if X is zero-dimensional or an interval.
And finally we show that for a Borel subset X of real numbers the product is CDH iff X
is a G-delta zero-dimensional set or an interval.
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Allen, Cristian Gerardo. A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers, dissertation, August 2017; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc1011753/m1/2/: accessed January 24, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .